Understanding Critical Points in Multivariable Functions

[erica1451]

Homework Statement

f(x,y,z)=(xy+yz+xz)/(1+x^2+y^2+z^2)
Explain why f has no absolute maximum or minimum. How about critical points?

Homework Equations

Hint: it is simplest to make 3 cases: a) x+y+z does not =0 b) x+y+z=0 c) x=y=z=0

The Attempt at a Solution

I did cases b and c, but I'm not sure how to go about doing a. Also, I'm not sure how to explain why the function does not have an absolute max or min.

 

[erica1451]

Can anyone help?

 

[tim_lou]

hmmm, simplifying things,

$$f(x,y,z)=\frac{1}{2}\cdot\left[\frac{(x+y+z)^2+1}{1+x^2+y^2+z^2}-1\right]$$
how can you bound f(x,y,z) from below? what about from above? can you make f arbitrarily big?
can you make f(x,y,z) arbitrary close to some values? try some additional cases, suppose x=y=z not equaling zero?

edit: additional hint: cylindrical coordinate.

 

[erica1451]

Thank you!